Clearly the the set of natural numbers has an infinite cardinality because the integers go on forever. We’ll start by looking at the the set of natural numbers.
NATURAL NUMBERS WHOLE NUMBERS INTEGERS RATIONAL NUMBERS HOW TO
Let’s try to take a look and see if we can figure out how to tell one kind of infinity from another: However, the reasons for this are a little more complex than it seems at first. So, which of these arguments is right? Well, they both make important and correct points, but in this case, the cardinality of the set of real numbers is in fact "bigger" than the cardinality of the set of integers, even though they are both infinite quantities. Yes, we could say, there are more real numbers than there are integers, but since both the number of real numbers and the number of integers is infinite, the cardinalities of these two sets is the same, because you cannot have one infinite quantity which is bigger than another! In other words, you could argue, adding something or multiplying something by infinity won't change that infinity. On the other hand, we could argue that adding more elements to an already infinite quantity of elements doesn't change that infinite quantity. So there must be more real numbers than integers, or in other words, the cardinality of the the set of real numbers must be a "bigger infinity" than the cardinality of the the set of integers, right?
If I asked you how many integers there are, you would have to say, “infinitely many.” But the the set of real numbers also goes on forever, and the the set of integers is a PROPER subset of the the set of real numbers. The answer, as it turns out, is “no.” So some infinities are “bigger” than others! Is the cardinality of the set of real numbers "bigger" than the cardinality of the set of integers?įor example, the the set of integers (Remember that the the set of integers includes all positive and negative whole numbers, including zero) goes on forever. If you start to think about infinities, you will eventually come up against the question: Infinity can be a very tricky concept, but a very interesting one. But you may have noticed that most of the sets we work with in mathematics: the set of integers, the set of rational numbers, the set of real numbers, etc., are NOT finite! These sets all have an INFINITE number of elements, so they are called INFINITE SETS. So far we have mostly focused our attention on finite sets (that is, sets which are not infinite).
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